Editing Wave (section)

=== Amplitude and modulation ===
{{Main|Amplitude modulation}}
{{See also|Frequency modulation|Phase modulation}}

[[File:Amplitudemodulation.gif|class=skin-invert-image|thumb|Amplitude modulation can be achieved through ''f''(''x'',''t'') = 1.00×sin(2π/0.10×(''x''−1.00×''t'')) and ''g''(''x'',''t'') = 1.00×sin(2π/0.11×(''x''−1.00×''t'')) only the resultant is visible to improve clarity of waveform.|left]]
[[File:Wave packet.svg|class=skin-invert-image|right|thumb|Illustration of the ''envelope'' (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the ''carrier'' wave, which is being modulated.]]

The amplitude of a wave may be constant (in which case the wave is a ''c.w.'' or ''[[continuous wave]]''), or may be ''modulated'' so as to vary with time and/or position. The outline of the variation in amplitude is called the ''envelope'' of the wave. Mathematically, the [[Amplitude modulation|modulated wave]] can be written in the form:<ref name=Jirauschek>{{cite book |url = https://books.google.com/books?id=6kOoT_AX2CwC&pg=PA9 |page = 9 |title = FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection |author = Christian Jirauschek |isbn = 978-3-86537-419-6 |year = 2005 |publisher = Cuvillier Verlag }}</ref><ref name="Kneubühl">{{cite book |title = Oscillations and waves |author = Fritz Kurt Kneubühl |url = https://books.google.com/books?id=geYKPFoLgoMC&pg=PA365 |page = 365 |year = 1997 |isbn = 978-3-540-62001-3 |publisher = Springer }}</ref><ref name=Lundstrom>{{cite book |url = https://books.google.com/books?id=FTdDMtpkSkIC&pg=PA33 |page = 33 |author = Mark Lundstrom |isbn = 978-0-521-63134-1 |year = 2000 |title = Fundamentals of carrier transport |publisher = Cambridge University Press }}</ref>
<math display="block">u(x,t) = A(x,t) \sin \left(kx - \omega t + \phi \right) , </math>
where <math>A(x,\ t)</math> is the amplitude envelope of the wave, <math>k</math> is the ''[[wavenumber]]'' and <math>\phi</math> is the ''[[phase (waves)|phase]]''. If the [[group velocity]] <math>v_g</math> (see below) is wavelength-independent, this equation can be simplified as:<ref name=Chen>{{cite book |chapter-url = https://books.google.com/books?id=LxzWPskhns0C&pg=PA363 |author = Chin-Lin Chen |title = Foundations for guided-wave optics |page = 363 |chapter = §13.7.3 Pulse envelope in nondispersive media |isbn = 978-0-471-75687-3 |year = 2006 |publisher = Wiley }}</ref>
<math display="block">u(x,t) = A(x - v_g t) \sin \left(kx - \omega t + \phi \right) , </math>
showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an ''envelope equation''.<ref name=Chen/><ref name="Recami">{{cite book |last1=Longhi |first1=Stefano |title=Localized Waves |last2=Janner |first2=Davide |publisher=Wiley-Interscience |year=2008 |isbn=978-0-470-10885-7 |editor1=Hugo E. Hernández-Figueroa |page=329 |chapter=Localization and Wannier wave packets in photonic crystals |editor2=Michel Zamboni-Rached |editor3=Erasmo Recami |chapter-url=https://books.google.com/books?id=xxbXgL967PwC&pg=PA329}}</ref>